{
  "id": "1303.1167",
  "version": "v1",
  "published": "2013-03-05T20:34:25.000Z",
  "updated": "2013-03-05T20:34:25.000Z",
  "title": "Invariance of Convex Sets for Non-autonomous Evolution Equations Governed by Forms",
  "authors": [
    "Wolfgang Arendt",
    "Dominik Dier",
    "El Maati Ouhabaz"
  ],
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We consider a non-autonomous form $\\fra:[0,T]\\times V\\times V \\to \\C$ where $V$ is a Hilbert space which is densely and continuously embedded in another Hilbert space $H$. Denote by $\\A(t) \\in \\L(V,V')$ the associated operator. Given $f \\in L^2(0,T, V')$, one knows that for each $u_0 \\in H$ there is a unique solution $u\\in H^1(0,T;V')\\cap L^2(0,T;V)$ of $$\\dot u(t) + \\A(t) u(t) = f(t), \\, \\, u(0) = u_0.$$ %\\begin{align*} %&\\dot u(t) + \\A(t)u(t)= f(t)\\ %& u(0)=u_0. %\\end{align*} This result by J. L. Lions is well-known. The aim of this article is to find a criterion for the invariance of a closed convex subset $\\Conv$ of $H$; i.e.\\ we give a criterion on the form which implies that $u(t)\\in \\Conv$ for all $t\\in[0,T]$ whenever $u_0\\in\\Conv$. In the autonomous case for $f = 0$, the criterion is known and even equivalent to invariance by a result proved in \\cite{Ouh96} (see also \\cite{Ouh05}). We give applications to positivity and comparison of solutions to heat equations with non-autonomous Robin boundary conditions. We also prove positivity of the solution to a quasi-linear heat equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-05T20:34:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-autonomous evolution equations",
      "convex sets",
      "invariance",
      "hilbert space",
      "non-autonomous robin boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.1167A"
    }
  }
}